This is a toy example of application of Bayesian Optimization. We will use the Bayesian Optimization to find the maximum value of one simple function.
### install.packages("GPfit")
library(GPfit)
library(dplyr)
library(ggplot2)
library(purrr)
We will optimize this simple function:
\[\left(6x-2\right)^2sin\left(12x-4\right)\]
First we define the function in R, then draw the graph of this function in the axes, and generate n0 initial points.
### Generate the function
f <- function(x) {
return((6*x-2)^2*sin(12*x-4))
}
### Draw the graph of this function
x <- seq(0, 1, by=0.0001)
y <- f(x)
graph_data <- data.frame(x, y)
graph_data %>% ggplot() +
geom_point(mapping = aes(x = x, y = y), size = 0.01, color = "navyblue") +
theme_bw()
y_min <- min(y)
min <- 0
for (i in x) {
if(f(i) < min) {
min <- f(i)
x_min <- i
}
}
### Generate the initial points
evaluation <- data.frame("x" = c(0, 1/3, 2/3, 1), "y" = f(c(0, 1/3, 2/3, 1)))
Here we build the initial GP model. We choose the power exponential correlation function here.
fit <- GP_fit(
X = evaluation[, "x"],
Y = evaluation[, "y"],
corr = list(type = "exponential", power = 1.95)
)
x_new <- seq(0, 1, length.out = 100)
pred <- predict.GP(fit, xnew = data.frame(x = x_new))
mu <- pred$Y_hat
sigma <- sqrt(pred$MSE)
pred <- pred %>% data.frame()
pred %>% ggplot() +
geom_line(mapping = aes(x = complete_data.xnew.1, y = Y_hat), color = "gold") +
geom_ribbon(mapping = aes(x = complete_data.xnew.1, y = Y_hat, xmin = 0, xmax = 1, ymin = Y_hat-sqrt(MSE), ymax = Y_hat+sqrt(MSE)), fill = "navyblue", alpha = 0.3) +
geom_point(mapping = aes(x = evaluation[1, "x"], y = evaluation[1, "y"]), color = "navyblue") +
geom_point(mapping = aes(x = evaluation[2, "x"], y = evaluation[2, "y"]), color = "navyblue") +
geom_point(mapping = aes(x = evaluation[3, "x"], y = evaluation[3, "y"]), color = "navyblue") +
geom_point(mapping = aes(x = evaluation[4, "x"], y = evaluation[4, "y"]), color = "navyblue") +
xlab("x") +
ylab("f(x)") +
theme_bw()
Then we will use diferent acquisition functions to find the x to evaluate next.
### y_best is the current best y value
y_best <- min(evaluation[, "y"])
### Make the f(x) - x plot a function
plot <- function(x_next) {
pred %>% ggplot() +
geom_line(mapping = aes(x = complete_data.xnew.1, y = Y_hat), color = "gold") +
geom_ribbon(mapping = aes(x = complete_data.xnew.1, y = Y_hat, xmin = 0, xmax = 1, ymin = Y_hat-sqrt(MSE), ymax = Y_hat+sqrt(MSE)), fill = "navyblue", alpha = 0.3) +
geom_point(mapping = aes(x = evaluation[1, "x"], y = evaluation[1, "y"]), color = "navyblue") +
geom_point(mapping = aes(x = evaluation[2, "x"], y = evaluation[2, "y"]), color = "navyblue") +
geom_point(mapping = aes(x = evaluation[3, "x"], y = evaluation[3, "y"]), color = "navyblue") +
geom_point(mapping = aes(x = evaluation[4, "x"], y = evaluation[4, "y"]), color = "navyblue") +
geom_vline(mapping = aes(xintercept = x_next), color = "red", linetype = "dashed") +
geom_point(mapping = aes(x = x_next, y = Y_hat[ceiling(x_next/0.01)]), color = "red") +
xlab("x") +
ylab("f(x)") +
theme_bw()
}
1.Probability of improvement
probability_improvement <- map2_dbl(
mu,
sigma,
function(m, s) {
if (s == 0) return(0)
else {
poi <- pnorm((y_best - m) / s)
# poi <- 1 - poi (if maximizing)
return(poi)
}
}
)
x_next_POI <- x_new[which.max(probability_improvement)]
ggplot() +
geom_line(mapping = aes(x = x_new, y = probability_improvement), color = "navyblue") +
geom_vline(mapping = aes(xintercept = x_next_POI), color = "red", linetype = "dashed") +
xlab("x") +
theme_bw()
plot(x_next_POI)
expected_improvement <- map2_dbl(
mu, sigma,
function(m, s) {
if (s == 0) return(0)
gamma <- (y_best - m) / s
phi <- pnorm(gamma)
return(s * (gamma * phi + dnorm(gamma)))
}
)
x_next_EI <- x_new[which.max(expected_improvement)]
ggplot() +
geom_line(mapping = aes(x = x_new, y = expected_improvement), color = "navyblue") +
geom_vline(mapping = aes(xintercept = x_next_EI), color = "red", linetype = "dashed") +
xlab("x") +
theme_bw()
plot(x_next_EI)
kappa <- 2 # tunable
lower_confidence_bound <- mu - kappa * sigma
# if maximizing: upper_confidence_bound <- mu + kappa * sigma
### Notice here is min, not max.
x_next_LCB <- x_new[which.min(lower_confidence_bound)]
ggplot() +
geom_line(mapping = aes(x = x_new, y = lower_confidence_bound), color = "navyblue") +
geom_vline(mapping = aes(xintercept = x_next_LCB), color = "red", linetype = "dashed") +
xlab("x") +
theme_bw()
plot(x_next_LCB)
In the last section, we discussed how Bayesian optimization works in one loop. In this section, we will continue looping until we got the optimized value of the simple function.
Firstly we will choose a “good” acquisition function.
This acquisition function measures the likelihood that the next x is better(lower or higher depends on our goal) than the current optimal value.
evaluation_t <- evaluation
t <- 1
data <- data.frame()
data$Y_hat <- c()
data$MSE <- c()
data$iteration <- c()
data$x_x <- c()
data$y_y <- c()
pred <- predict.GP(fit, xnew = data.frame(x = x_new))
pred <- pred %>% data.frame() %>% select(c("Y_hat", "MSE", "complete_data.xnew.1"))
pred <- pred %>% mutate("iteration" = 0) %>% mutate("x_x" = NA) %>% mutate("y_y" = NA)
data <- rbind(data, pred)
acq_data = data.frame()
probability_improvement <- probability_improvement %>%
as.data.frame() %>%
mutate("iteration" = 1) %>%
mutate("x_next_POI" = NA)
colnames(probability_improvement) <- c("probability_improvement", "iteration", "x_next_POI")
acq_data <- rbind(acq_data, probability_improvement)
x_next_df_POI <- data.frame(x_next_POI) %>% mutate("probability_improvement" = NA) %>% mutate("iteration" = 1)
colnames(x_next_df_POI) <- c("x_next_POI", "probability_improvement", "iteration")
acq_data <- rbind(acq_data, x_next_df_POI)
while(t <= 10) {
evaluation[nrow(evaluation)+1,] = c(x_next_POI, f(x_next_POI))
fit <- GP_fit(
X = evaluation[, "x"],
Y = evaluation[, "y"],
corr = list(type = "exponential", power = 1.95)
)
x_new <- seq(0, 1, length.out = 100)
pred <- predict.GP(fit, xnew = data.frame(x = x_new))
mu <- pred$Y_hat
sigma <- sqrt(pred$MSE)
pred <- pred %>% data.frame() %>% select(c("Y_hat", "MSE", "complete_data.xnew.1"))
pred <- pred %>% mutate("iteration" = t) %>% mutate("x_x" = NA) %>% mutate("y_y" = NA)
data <- rbind(data, pred)
probability_improvement <- map2_dbl(
mu,
sigma,
function(m, s) {
if (s == 0) return(0)
else {
poi <- pnorm((y_best - m) / s)
# poi <- 1 - poi (if maximizing)
return(poi)
}
}
)
x_next_POI <- x_new[which.max(probability_improvement)]
probability_improvement <- probability_improvement %>% as.data.frame()
colnames(probability_improvement) <- "probability_improvement"
probability_improvement <- probability_improvement %>%
mutate("iteration" = t+1) %>%
mutate("x_next_POI" = NA)
acq_data <- rbind(acq_data, probability_improvement)
x_next_df <- data.frame(x_next_POI) %>% mutate("probability_improvement" = NA) %>% mutate("iteration" = t+1)
acq_data <- rbind(acq_data, x_next_df)
t <- t+1
}
t <- 1
nr <- nrow(data)
for (i in 4:14) {
for (j in 1:i) {
data[nr + t,] = c(NA, NA, NA, i-4, evaluation[j, 1], evaluation[j, 2])
t <- t+1
}
}
plot_1 <- data %>% ggplot() +
geom_line(mapping = aes(x = complete_data.xnew.1, y = Y_hat), color = "gold") +
geom_ribbon(mapping = aes(x = complete_data.xnew.1, y = Y_hat, xmin = 0, xmax = 1, ymin = Y_hat-sqrt(MSE), ymax = Y_hat+sqrt(MSE)), fill = "navyblue", alpha = 0.3) +
geom_point(mapping = aes(x = x_x, y = y_y)) +
facet_wrap(~iteration) +
xlab("x") +
ylab("f(x)") +
theme_bw()
x_new_df <- rep(c(x_new, NA),10) %>% as.data.frame()
colnames(x_new_df) <- "x_new"
acq_data <- cbind(acq_data %>% filter(iteration<11), x_new_df)
plot_2 <- acq_data %>% ggplot() +
geom_line(mapping = aes(x = x_new, y = probability_improvement), color = "navyblue") +
geom_vline(mapping = aes(xintercept = x_next_POI), color = "red", linetype = "dashed") +
xlab("x") +
facet_wrap(~ iteration) +
theme_bw()
print(plot_1)
print(plot_2)
This acquisition function takes how large the improvement is into account.
evaluation <- evaluation_t
t <- 1
data <- data.frame()
data$Y_hat <- c()
data$MSE <- c()
data$iteration <- c()
data$x_x <- c()
data$y_y <- c()
pred <- predict.GP(fit, xnew = data.frame(x = x_new))
pred <- pred %>% data.frame() %>% select(c("Y_hat", "MSE", "complete_data.xnew.1"))
pred <- pred %>% mutate("iteration" = 0) %>% mutate("x_x" = NA) %>% mutate("y_y" = NA)
data <- rbind(data, pred)
acq_data = data.frame()
expected_improvement <- expected_improvement %>%
as.data.frame() %>%
mutate("iteration" = 1) %>%
mutate("x_next_EI" = NA)
colnames(expected_improvement) <- c("expected_improvement", "iteration", "x_next_EI")
acq_data <- rbind(acq_data, expected_improvement)
x_next_df_EI <- data.frame(x_next_EI) %>% mutate("expected_improvement" = NA) %>% mutate("iteration" = 1)
colnames(x_next_df_EI) <- c("x_next_EI", "expected_improvement", "iteration")
acq_data <- rbind(acq_data, x_next_df_EI)
while(t <= 10) {
evaluation[nrow(evaluation)+1,] = c(x_next_EI, f(x_next_EI))
fit <- GP_fit(
X = evaluation[, "x"],
Y = evaluation[, "y"],
corr = list(type = "exponential", power = 1.95)
)
x_new <- seq(0, 1, length.out = 100)
pred <- predict.GP(fit, xnew = data.frame(x = x_new))
mu <- pred$Y_hat
sigma <- sqrt(pred$MSE)
pred <- pred %>% data.frame() %>% select(c("Y_hat", "MSE", "complete_data.xnew.1"))
pred <- pred %>% mutate("iteration" = t) %>% mutate("x_x" = NA) %>% mutate("y_y" = NA)
data <- rbind(data, pred)
expected_improvement <- map2_dbl(
mu, sigma,
function(m, s) {
if (s == 0) return(0)
gamma <- (y_best - m) / s
phi <- pnorm(gamma)
return(s * (gamma * phi + dnorm(gamma)))
}
)
x_next_EI <- x_new[which.max(expected_improvement)]
expected_improvement <- expected_improvement %>% as.data.frame()
colnames(expected_improvement) <- "expected_improvement"
expected_improvement <- expected_improvement %>%
mutate("iteration" = t+1) %>%
mutate("x_next_EI" = NA)
acq_data <- rbind(acq_data, expected_improvement)
x_next_df <- data.frame(x_next_EI) %>% mutate("expected_improvement" = NA) %>% mutate("iteration" = t+1)
acq_data <- rbind(acq_data, x_next_df)
t <- t+1
}
t <- 1
nr <- nrow(data)
for (i in 4:14) {
for (j in 1:i) {
data[nr + t,] = c(NA, NA, NA, i-4, evaluation[j, 1], evaluation[j, 2])
t <- t+1
}
}
plot_1 <- data %>% ggplot() +
geom_line(mapping = aes(x = complete_data.xnew.1, y = Y_hat), color = "gold") +
geom_ribbon(mapping = aes(x = complete_data.xnew.1, y = Y_hat, xmin = 0, xmax = 1, ymin = Y_hat-sqrt(MSE), ymax = Y_hat+sqrt(MSE)), fill = "navyblue", alpha = 0.3) +
geom_point(mapping = aes(x = x_x, y = y_y)) +
facet_wrap(~iteration) +
xlab("x") +
ylab("f(x)") +
theme_bw()
x_new_df <- rep(c(x_new, NA),10) %>% as.data.frame()
colnames(x_new_df) <- "x_new"
acq_data <- cbind(acq_data %>% filter(iteration < 11), x_new_df)
plot_2 <- acq_data %>% ggplot() +
geom_line(mapping = aes(x = x_new, y = expected_improvement), color = "navyblue") +
geom_vline(mapping = aes(xintercept = x_next_EI), color = "red", linetype = "dashed") +
xlab("x") +
facet_wrap(~ iteration) +
theme_bw()
print(plot_1)
print(plot_2)
### Here we save the data to use it later.
data_EI <- data
This acquisition functions balances the area where mean(x) is large and the area where sigma(x) is large. In other words, this acquisition function trades off between exploration and exploitation.
evaluation <- evaluation_t
t <- 1
data <- data.frame()
data$Y_hat <- c()
data$MSE <- c()
data$iteration <- c()
data$x_x <- c()
data$y_y <- c()
pred <- predict.GP(fit, xnew = data.frame(x = x_new))
pred <- pred %>% data.frame() %>% select(c("Y_hat", "MSE", "complete_data.xnew.1"))
pred <- pred %>% mutate("iteration" = 0) %>% mutate("x_x" = NA) %>% mutate("y_y" = NA)
data <- rbind(data, pred)
acq_data = data.frame()
lower_confidence_bound <- lower_confidence_bound %>%
as.data.frame() %>%
mutate("iteration" = 1) %>%
mutate("x_next_LCB" = NA)
colnames(lower_confidence_bound) <- c("lower_confidence_bound", "iteration", "x_next_LCB")
acq_data <- rbind(acq_data, lower_confidence_bound)
x_next_df_LCB <- data.frame(x_next_LCB) %>% mutate("lower_confidence_bound" = NA) %>% mutate("iteration" = 1)
colnames(x_next_df_LCB) <- c("x_next_LCB", "lower_confidence_bound", "iteration")
acq_data <- rbind(acq_data, x_next_df_LCB)
while(t <= 8) {
evaluation[nrow(evaluation)+1,] = c(x_next_LCB, f(x_next_LCB))
fit <- GP_fit(
X = evaluation[, "x"],
Y = evaluation[, "y"],
corr = list(type = "exponential", power = 1.95)
)
x_new <- seq(0, 1, length.out = 100)
pred <- predict.GP(fit, xnew = data.frame(x = x_new))
mu <- pred$Y_hat
sigma <- sqrt(pred$MSE)
pred <- pred %>% data.frame() %>% select(c("Y_hat", "MSE", "complete_data.xnew.1"))
pred <- pred %>% mutate("iteration" = t) %>% mutate("x_x" = NA) %>% mutate("y_y" = NA)
data <- rbind(data, pred)
lower_confidence_bound <- mu - kappa * sigma
x_next_LCB <- x_new[which.min(lower_confidence_bound)]
lower_confidence_bound <- lower_confidence_bound %>% as.data.frame()
colnames(lower_confidence_bound) <- "lower_confidence_bound"
lower_confidence_bound <- lower_confidence_bound %>%
mutate("iteration" = t+1) %>%
mutate("x_next_LCB" = NA)
acq_data <- rbind(acq_data, lower_confidence_bound)
x_next_df <- data.frame(x_next_LCB) %>% mutate("lower_confidence_bound" = NA) %>% mutate("iteration" = t+1)
acq_data <- rbind(acq_data, x_next_df)
t <- t+1
}
t <- 1
nr <- nrow(data)
for (i in 4:12) {
for (j in 1:i) {
data[nr + t,] = c(NA, NA, NA, i-4, evaluation[j, 1], evaluation[j, 2])
t <- t+1
}
}
plot_1 <- data %>% ggplot() +
geom_line(mapping = aes(x = complete_data.xnew.1, y = Y_hat), color = "gold") +
geom_ribbon(mapping = aes(x = complete_data.xnew.1, y = Y_hat, xmin = 0, xmax = 1, ymin = Y_hat-sqrt(MSE), ymax = Y_hat+sqrt(MSE)), fill = "navyblue", alpha = 0.3) +
geom_point(mapping = aes(x = x_x, y = y_y)) +
facet_wrap(~iteration) +
xlab("x") +
ylab("f(x)") +
theme_bw()
x_new_df <- rep(c(x_new, NA),8) %>% as.data.frame()
colnames(x_new_df) <- "x_new"
acq_data <- cbind(acq_data %>% filter(iteration < 9), x_new_df)
plot_2 <- acq_data %>% ggplot() +
geom_line(mapping = aes(x = x_new, y = lower_confidence_bound), color = "navyblue") +
geom_vline(mapping = aes(xintercept = x_next_LCB), color = "red", linetype = "dashed") +
xlab("x") +
facet_wrap(~ iteration) +
theme_bw()
print(plot_1)
print(plot_2)
Here we give a detailed example to show the Expected Improvement acqusition function.
t <- 1
while(t < 10) {
print(data_EI %>% filter(iteration == t) %>% ggplot() +
geom_line(mapping = aes(x = complete_data.xnew.1, y = Y_hat), color = "gold") +
geom_ribbon(mapping = aes(x = complete_data.xnew.1, y = Y_hat, xmin = 0, xmax = 1, ymin = Y_hat-sqrt(MSE), ymax = Y_hat+sqrt(MSE)), fill = "navyblue", alpha = 0.3) +
geom_point(mapping = aes(x = x_x, y = y_y)) +
xlab("x") +
ylab("f(x)") +
labs(paste("The", t, "iteration")) +
theme_bw())
t <- t+1
}
Then we pick the x value that has the minimal f(x) value from what we evaluated.
### Pick the x that has the minimal f(x) value from what we evaluated.
cat("The opmized x we find is", evaluation[which.min(evaluation$y),]$x)
## The opmized x we find is 0.7575758
### Compare it with the real x that minimize f(x) in [0,1]
cat("The real optimized x is", x_min)
## The real optimized x is 0.7572
We can see that we use 10 evaluations to get the x, and the result is pretty close to the real one.
In this section, we will compare the Bayesian optimization with quasi-Newton method. We will try several initial guesses.
### Firstly, we build several equally spaced initial guesses in [0, 1].
ini_guess <- seq(0, 1, length.out = 9)
### Then we will use quasi-Newton method to try to find the optimal f(x) value.
n=1
for(t in ini_guess) {
res <- optim(t, f, method = "BFGS", lower = 0, upper = 1)
writeLines(paste("The", n, "initial guess x0 = ", t, ":\nx=", res$par,"\nf(x)=", res$value, "\nTimes of iteration=", res$counts[1], "\n-----------------------------------------------"))
n <- n+1
}
## The 1 initial guess x0 = 0 :
## x= 0.142589801857248
## f(x)= -0.986325406263923
## Times of iteration= 49
## -----------------------------------------------
## The 2 initial guess x0 = 0.125 :
## x= 0.142591747065892
## f(x)= -0.986325405324459
## Times of iteration= 19
## -----------------------------------------------
## The 3 initial guess x0 = 0.25 :
## x= 0.142588440215961
## f(x)= -0.986325406235592
## Times of iteration= 10
## -----------------------------------------------
## The 4 initial guess x0 = 0.375 :
## x= 0.142591304993005
## f(x)= -0.986325405639193
## Times of iteration= 19
## -----------------------------------------------
## The 5 initial guess x0 = 0.5 :
## x= 0.142589561542382
## f(x)= -0.986325406299975
## Times of iteration= 47
## -----------------------------------------------
## The 6 initial guess x0 = 0.625 :
## x= 0.757247331284393
## f(x)= -6.02074005468035
## Times of iteration= 9
## -----------------------------------------------
## The 7 initial guess x0 = 0.75 :
## x= 0.757247332200751
## f(x)= -6.02074005468175
## Times of iteration= 6
## -----------------------------------------------
## The 8 initial guess x0 = 0.875 :
## x= 0.757247564706613
## f(x)= -6.02074005500689
## Times of iteration= 26
## -----------------------------------------------
## The 9 initial guess x0 = 1 :
## x= 0.75724733297612
## f(x)= -6.02074005468293
## Times of iteration= 10
## -----------------------------------------------
We can see there are two disadvantages of quasi-Newton method. The first one is we have to choose a “good” initial guess to find the global optimum. In our example, if the initial guess is bigger than 0.5, the quasi-Newton method will find a local optimum.
The second one is it will need more times of iteration to get the optimal value. The minimum number of iteration in our example is 13 times, which is 3 times more than our Bayesian optimization example.
Also, because here we use a simple function. If we want to optimize a complex function, which cannot or very difficult to get the derivatives, the quasi-Newton method cannot or hard to find the optimum, but the Bayesian optimization can do that.
In the previous part, we use 4 evenly spaced initial points. In this part, we will try to use different sets of initial points.
Firstly, we will try 3 evenly spaced points between 0 and 1 as the initial set of points.
evaluation <- data.frame("x" = c(0, 1/2, 1), "y" = f(c(0, 1/2, 1)))
pip <- function(evaluation, num) {
fit <- GP_fit(
X = evaluation[, "x"],
Y = evaluation[, "y"],
corr = list(type = "exponential", power = 1.95)
)
t <- 1
data <- data.frame()
data$Y_hat <- c()
data$MSE <- c()
data$iteration <- c()
data$x_x <- c()
data$y_y <- c()
pred <- predict.GP(fit, xnew = data.frame(x = x_new))
pred <- pred %>% data.frame() %>% select(c("Y_hat", "MSE", "complete_data.xnew.1"))
pred <- pred %>% mutate("iteration" = 0) %>% mutate("x_x" = NA) %>% mutate("y_y" = NA)
data <- rbind(data, pred)
acq_data = data.frame()
expected_improvement <- expected_improvement %>%
as.data.frame() %>%
mutate("iteration" = 1) %>%
mutate("x_next_EI" = NA)
colnames(expected_improvement) <- c("expected_improvement", "iteration", "x_next_EI")
acq_data <- rbind(acq_data, expected_improvement)
x_next_df_EI <- data.frame(x_next_EI) %>% mutate("expected_improvement" = NA) %>% mutate("iteration" = 1)
colnames(x_next_df_EI) <- c("x_next_EI", "expected_improvement", "iteration")
acq_data <- rbind(acq_data, x_next_df_EI)
while(t <= 10) {
evaluation[nrow(evaluation)+1,] = c(x_next_EI, f(x_next_EI))
fit <- GP_fit(
X = evaluation[, "x"],
Y = evaluation[, "y"],
corr = list(type = "exponential", power = 1.95)
)
x_new <- seq(0, 1, length.out = 100)
pred <- predict.GP(fit, xnew = data.frame(x = x_new))
mu <- pred$Y_hat
sigma <- sqrt(pred$MSE)
pred <- pred %>% data.frame() %>% select(c("Y_hat", "MSE", "complete_data.xnew.1"))
pred <- pred %>% mutate("iteration" = t) %>% mutate("x_x" = NA) %>% mutate("y_y" = NA)
data <- rbind(data, pred)
expected_improvement <- map2_dbl(
mu, sigma,
function(m, s) {
if (s == 0) return(0)
gamma <- (y_best - m) / s
phi <- pnorm(gamma)
return(s * (gamma * phi + dnorm(gamma)))
}
)
x_next_EI <- x_new[which.max(expected_improvement)]
expected_improvement <- expected_improvement %>% as.data.frame()
colnames(expected_improvement) <- "expected_improvement"
expected_improvement <- expected_improvement %>%
mutate("iteration" = t+1) %>%
mutate("x_next_EI" = NA)
acq_data <- rbind(acq_data, expected_improvement)
x_next_df <- data.frame(x_next_EI) %>% mutate("expected_improvement" = NA) %>% mutate("iteration" = t+1)
acq_data <- rbind(acq_data, x_next_df)
t <- t+1
}
t <- 1
nr <- nrow(data)
for (i in (num):(num+10)) {
for (j in 1:i) {
data[nr + t,] = c(NA, NA, NA, i-num, evaluation[j, 1], evaluation[j, 2])
t <- t+1
}
}
plot_1 <- data %>% ggplot() +
geom_line(mapping = aes(x = complete_data.xnew.1, y = Y_hat), color = "gold") +
geom_ribbon(mapping = aes(x = complete_data.xnew.1, y = Y_hat, xmin = 0, xmax = 1, ymin = Y_hat-sqrt(MSE), ymax = Y_hat+sqrt(MSE)), fill = "navyblue", alpha = 0.3) +
geom_point(mapping = aes(x = x_x, y = y_y)) +
facet_wrap(~iteration) +
xlab("x") +
ylab("f(x)") +
theme_bw()
x_new_df <- rep(c(x_new, NA),10) %>% as.data.frame()
colnames(x_new_df) <- "x_new"
acq_data <- cbind(acq_data %>% filter(iteration < 11), x_new_df)
plot_2 <- acq_data %>% ggplot() +
geom_line(mapping = aes(x = x_new, y = expected_improvement), color = "navyblue") +
geom_vline(mapping = aes(xintercept = x_next_EI), color = "red", linetype = "dashed") +
xlab("x") +
facet_wrap(~ iteration) +
theme_bw()
print(plot_1)
print(plot_2)
return(data)
}
data <- pip(evaluation, 3)
Here we use four randomly located points instead of four evenly spaced points as the initial points set.
set.seed(1234)
x <- sample.int(100,4)/100
evaluation <- data.frame("x" = x, "y" = f(x))
pip(evaluation, 4)
## Y_hat MSE complete_data.xnew.1 iteration x_x
## 1 -0.314345885 4.486900e-01 0.00000000 0 NA
## 2 -0.321626382 3.411207e-01 0.01010101 0 NA
## 3 -0.332389212 2.504301e-01 0.02020202 0 NA
## 4 -0.346264928 1.758942e-01 0.03030303 0 NA
## 5 -0.362836383 1.165089e-01 0.04040404 0 NA
## 6 -0.381650842 7.103962e-02 0.05050505 0 NA
## 7 -0.402237945 3.808478e-02 0.06060606 0 NA
## 8 -0.424139380 1.615572e-02 0.07070707 0 NA
## 9 -0.446971415 3.800753e-03 0.08080808 0 NA
## 10 -0.470699024 4.875145e-05 0.09090909 0 NA
## 11 -0.495599231 4.080030e-03 0.10101010 0 NA
## 12 -0.519572139 1.113635e-02 0.11111111 0 NA
## 13 -0.541212854 1.848639e-02 0.12121212 0 NA
## 14 -0.559413819 2.446129e-02 0.13131313 0 NA
## 15 -0.573211515 2.809099e-02 0.14141414 0 NA
## 16 -0.581756323 2.897411e-02 0.15151515 0 NA
## 17 -0.584304437 2.718049e-02 0.16161616 0 NA
## 18 -0.580213586 2.315554e-02 0.17171717 0 NA
## 19 -0.568934346 1.762201e-02 0.18181818 0 NA
## 20 -0.549987977 1.148311e-02 0.19191919 0 NA
## 21 -0.522910715 5.738708e-03 0.20202020 0 NA
## 22 -0.487092216 1.450842e-03 0.21212121 0 NA
## 23 -0.440789158 1.541379e-04 0.22222222 0 NA
## 24 -0.382751000 2.292146e-03 0.23232323 0 NA
## 25 -0.317738694 4.427089e-03 0.24242424 0 NA
## 26 -0.248647321 5.043925e-03 0.25252525 0 NA
## 27 -0.178015022 3.722501e-03 0.26262626 0 NA
## 28 -0.108519663 1.194379e-03 0.27272727 0 NA
## 29 -0.044006518 2.930805e-04 0.28282828 0 NA
## 30 0.012508779 4.886718e-03 0.29292929 0 NA
## 31 0.062616567 1.518785e-02 0.30303030 0 NA
## 32 0.106184167 3.219325e-02 0.31313131 0 NA
## 33 0.142674750 5.707815e-02 0.32323232 0 NA
## 34 0.171439253 9.102767e-02 0.33333333 0 NA
## 35 0.191809531 1.351565e-01 0.34343434 0 NA
## 36 0.203143616 1.904431e-01 0.35353535 0 NA
## 37 0.204852626 2.576687e-01 0.36363636 0 NA
## 38 0.196418582 3.373622e-01 0.37373737 0 NA
## 39 0.177406799 4.297508e-01 0.38383838 0 NA
## 40 0.147474532 5.347215e-01 0.39393939 0 NA
## 41 0.106376726 6.517921e-01 0.40404040 0 NA
## 42 0.053969401 7.800968e-01 0.41414141 0 NA
## 43 -0.009788991 9.183848e-01 0.42424242 0 NA
## 44 -0.084837977 1.065034e+00 0.43434343 0 NA
## 45 -0.171017008 1.218076e+00 0.44444444 0 NA
## 46 -0.268068741 1.375242e+00 0.45454545 0 NA
## 47 -0.375643449 1.534008e+00 0.46464646 0 NA
## 48 -0.493304372 1.691662e+00 0.47474747 0 NA
## 49 -0.620533828 1.845369e+00 0.48484848 0 NA
## 50 -0.756739899 1.992246e+00 0.49494949 0 NA
## 51 -0.901263537 2.129436e+00 0.50505051 0 NA
## 52 -1.053385923 2.254187e+00 0.51515152 0 NA
## 53 -1.212335967 2.363918e+00 0.52525253 0 NA
## 54 -1.377297797 2.456292e+00 0.53535354 0 NA
## 55 -1.547418171 2.529272e+00 0.54545455 0 NA
## 56 -1.721813707 2.581174e+00 0.55555556 0 NA
## 57 -1.899577876 2.610713e+00 0.56565657 0 NA
## 58 -2.079787726 2.617030e+00 0.57575758 0 NA
## 59 -2.261510290 2.599721e+00 0.58585859 0 NA
## 60 -2.443808690 2.558844e+00 0.59595960 0 NA
## 61 -2.625747923 2.494921e+00 0.60606061 0 NA
## 62 -2.806400343 2.408933e+00 0.61616162 0 NA
## 63 -2.984850873 2.302299e+00 0.62626263 0 NA
## 64 -3.160201948 2.176853e+00 0.63636364 0 NA
## 65 -3.331578234 2.034809e+00 0.64646465 0 NA
## 66 -3.498131135 1.878717e+00 0.65656566 0 NA
## 67 -3.659043103 1.711424e+00 0.66666667 0 NA
## 68 -3.813531778 1.536015e+00 0.67676768 0 NA
## 69 -3.960853925 1.355761e+00 0.68686869 0 NA
## 70 -4.100309181 1.174061e+00 0.69696970 0 NA
## 71 -4.231243556 9.943801e-01 0.70707071 0 NA
## 72 -4.353052630 8.201909e-01 0.71717172 0 NA
## 73 -4.465184336 6.549128e-01 0.72727273 0 NA
## 74 -4.567141163 5.018553e-01 0.73737374 0 NA
## 75 -4.658481493 3.641650e-01 0.74747475 0 NA
## 76 -4.738819531 2.447799e-01 0.75757576 0 NA
## 77 -4.807822759 1.463944e-01 0.76767677 0 NA
## 78 -4.865204224 7.144482e-02 0.77777778 0 NA
## 79 -4.910701243 2.214052e-02 0.78787879 0 NA
## 80 -4.943995174 6.783098e-04 0.79797980 0 NA
## 81 -4.964403447 1.015011e-02 0.80808081 0 NA
## 82 -4.972591614 4.934522e-02 0.81818182 0 NA
## 83 -4.969319582 1.165216e-01 0.82828283 0 NA
## 84 -4.955143056 2.103431e-01 0.83838384 0 NA
## 85 -4.930611929 3.293095e-01 0.84848485 0 NA
## 86 -4.896299930 4.716944e-01 0.85858586 0 NA
## 87 -4.852812003 6.355529e-01 0.86868687 0 NA
## 88 -4.800784425 8.187520e-01 0.87878788 0 NA
## 89 -4.740881754 1.019011e+00 0.88888889 0 NA
## 90 -4.673792158 1.233949e+00 0.89898990 0 NA
## 91 -4.600221865 1.461125e+00 0.90909091 0 NA
## 92 -4.520889139 1.698092e+00 0.91919192 0 NA
## 93 -4.436518036 1.942433e+00 0.92929293 0 NA
## 94 -4.347832147 2.191802e+00 0.93939394 0 NA
## 95 -4.255548442 2.443961e+00 0.94949495 0 NA
## 96 -4.160371353 2.696804e+00 0.95959596 0 NA
## 97 -4.062987188 2.948388e+00 0.96969697 0 NA
## 98 -3.964058945 3.196944e+00 0.97979798 0 NA
## 99 -3.864221595 3.440896e+00 0.98989899 0 NA
## 100 -3.764077898 3.678864e+00 1.00000000 0 NA
## 101 -0.444948730 1.015665e+00 0.00000000 1 NA
## 102 -0.425177012 7.796739e-01 0.01010101 1 NA
## 103 -0.411840762 5.764428e-01 0.02020202 1 NA
## 104 -0.404784852 4.065407e-01 0.03030303 1 NA
## 105 -0.403716377 2.694843e-01 0.04040404 1 NA
## 106 -0.408212466 1.637939e-01 0.05050505 1 NA
## 107 -0.417737294 8.711779e-02 0.06060606 1 NA
## 108 -0.431674611 3.643189e-02 0.07070707 1 NA
## 109 -0.449397907 8.357952e-03 0.08080808 1 NA
## 110 -0.470565425 1.014904e-04 0.09090909 1 NA
## 111 -0.495088350 8.999078e-03 0.10101010 1 NA
## 112 -0.520316620 2.502527e-02 0.11111111 1 NA
## 113 -0.544294888 4.191722e-02 0.12121212 1 NA
## 114 -0.565373425 5.562684e-02 0.13131313 1 NA
## 115 -0.582073817 6.375707e-02 0.14141414 1 NA
## 116 -0.593083388 6.535059e-02 0.15151515 1 NA
## 117 -0.597271731 6.067386e-02 0.16161616 1 NA
## 118 -0.593709700 5.095315e-02 0.17171717 1 NA
## 119 -0.581681949 3.807178e-02 0.18181818 1 NA
## 120 -0.560683981 2.425682e-02 0.19191919 1 NA
## 121 -0.530385217 1.179775e-02 0.20202020 1 NA
## 122 -0.490493423 2.883509e-03 0.21212121 1 NA
## 123 -0.439879355 2.973896e-04 0.22222222 1 NA
## 124 -0.378039557 4.542770e-03 0.23232323 1 NA
## 125 -0.310023742 8.875406e-03 0.24242424 1 NA
## 126 -0.239355545 1.015337e-02 0.25252525 1 NA
## 127 -0.169345824 7.470760e-03 0.26262626 1 NA
## 128 -0.103514030 2.365253e-03 0.27272727 1 NA
## 129 -0.046569024 5.737734e-04 0.28282828 1 NA
## 130 -0.002351282 9.942236e-03 0.29292929 1 NA
## 131 0.029995968 3.173334e-02 0.30303030 1 NA
## 132 0.049678681 6.871041e-02 0.31313131 1 NA
## 133 0.055585508 1.239010e-01 0.32323232 1 NA
## 134 0.046592237 2.001568e-01 0.33333333 1 NA
## 135 0.021664875 2.998768e-01 0.34343434 1 NA
## 136 -0.020087926 4.247699e-01 0.35353535 1 NA
## 137 -0.079383644 5.756534e-01 0.36363636 1 NA
## 138 -0.156746475 7.522970e-01 0.37373737 1 NA
## 139 -0.252495817 9.533220e-01 0.38383838 1 NA
## 140 -0.366740628 1.176163e+00 0.39393939 1 NA
## 141 -0.499378335 1.417096e+00 0.40404040 1 NA
## 142 -0.650097351 1.671329e+00 0.41414141 1 NA
## 143 -0.818382359 1.933157e+00 0.42424242 1 NA
## 144 -1.003521687 2.196158e+00 0.43434343 1 NA
## 145 -1.204616146 2.453437e+00 0.44444444 1 NA
## 146 -1.420588840 2.697890e+00 0.45454545 1 NA
## 147 -1.650195528 2.922483e+00 0.46464646 1 NA
## 148 -1.892035271 3.120532e+00 0.47474747 1 NA
## 149 -2.144561170 3.285966e+00 0.48484848 1 NA
## 150 -2.406091175 3.413577e+00 0.49494949 1 NA
## 151 -2.674819010 3.499222e+00 0.50505051 1 NA
## 152 -2.948825395 3.540001e+00 0.51515152 1 NA
## 153 -3.226089805 3.534381e+00 0.52525253 1 NA
## 154 -3.504503067 3.482267e+00 0.53535354 1 NA
## 155 -3.781881138 3.385026e+00 0.54545455 1 NA
## 156 -4.055980399 3.245446e+00 0.55555556 1 NA
## 157 -4.324514756 3.067651e+00 0.56565657 1 NA
## 158 -4.585174798 2.856949e+00 0.57575758 1 NA
## 159 -4.835649169 2.619634e+00 0.58585859 1 NA
## 160 -5.073648168 2.362747e+00 0.59595960 1 NA
## 161 -5.296929505 2.093787e+00 0.60606061 1 NA
## 162 -5.503325944 1.820413e+00 0.61616162 1 NA
## 163 -5.690774434 1.550119e+00 0.62626263 1 NA
## 164 -5.857346160 1.289915e+00 0.63636364 1 NA
## 165 -6.001276769 1.046037e+00 0.64646465 1 NA
## 166 -6.120995887 8.236791e-01 0.65656566 1 NA
## 167 -6.215154827 6.267893e-01 0.66666667 1 NA
## 168 -6.282651213 4.579336e-01 0.67676768 1 NA
## 169 -6.322648887 3.182359e-01 0.68686869 1 NA
## 170 -6.334590917 2.074088e-01 0.69696970 1 NA
## 171 -6.318202223 1.238719e-01 0.70707071 1 NA
## 172 -6.273475192 6.495760e-02 0.71717172 1 NA
## 173 -6.200621996 2.719951e-02 0.72727273 1 NA
## 174 -6.099938772 6.719906e-03 0.73737374 1 NA
## 175 -5.971199355 1.078762e-14 0.74747475 1 NA
## 176 -5.812933721 4.002599e-03 0.75757576 1 NA
## 177 -5.631362198 8.588462e-03 0.76767677 1 NA
## 178 -5.431430460 9.091875e-03 0.77777778 1 NA
## 179 -5.217539116 5.113544e-03 0.78787879 1 NA
## 180 -4.994355514 2.983417e-04 0.79797980 1 NA
## 181 -4.767926568 4.403168e-03 0.80808081 1 NA
## 182 -4.539828962 2.220251e-02 0.81818182 1 NA
## 183 -4.311409156 5.713974e-02 0.82828283 1 NA
## 184 -4.084768956 1.136539e-01 0.83838384 1 NA
## 185 -3.862048961 1.960721e-01 0.84848485 1 NA
## 186 -3.645284006 3.082074e-01 0.85858586 1 NA
## 187 -3.436335679 4.530611e-01 0.86868687 1 NA
## 188 -3.236852649 6.325994e-01 0.87878788 1 NA
## 189 -3.048246001 8.476075e-01 0.88888889 1 NA
## 190 -2.871674955 1.097624e+00 0.89898990 1 NA
## 191 -2.708040852 1.380953e+00 0.90909091 1 NA
## 192 -2.557988155 1.694751e+00 0.91919192 1 NA
## 193 -2.421911524 2.035175e+00 0.92929293 1 NA
## 194 -2.299968093 2.397574e+00 0.93939394 1 NA
## 195 -2.192094139 2.776721e+00 0.94949495 1 NA
## 196 -2.098025272 3.167050e+00 0.95959596 1 NA
## 197 -2.017319332 3.562901e+00 0.96969697 1 NA
## 198 -1.949381125 3.958738e+00 0.97979798 1 NA
## 199 -1.893488223 4.349353e+00 0.98989899 1 NA
## 200 -1.848817052 4.730028e+00 1.00000000 1 NA
## 201 -1.129314188 3.288345e+00 0.00000000 2 NA
## 202 -1.027461292 2.839801e+00 0.01010101 2 NA
## 203 -0.924943727 2.354229e+00 0.02020202 2 NA
## 204 -0.825081197 1.851290e+00 0.03030303 2 NA
## 205 -0.731436585 1.357009e+00 0.04040404 2 NA
## 206 -0.647583603 9.015996e-01 0.05050505 2 NA
## 207 -0.576842685 5.158754e-01 0.06060606 2 NA
## 208 -0.522015564 2.266928e-01 0.07070707 2 NA
## 209 -0.485174989 5.236793e-02 0.08080808 2 NA
## 210 -0.467789556 5.621313e-04 0.09090909 2 NA
## 211 -0.470433394 6.459775e-02 0.10101010 2 NA
## 212 -0.488826889 2.015562e-01 0.11111111 2 NA
## 213 -0.518437353 3.659769e-01 0.12121212 2 NA
## 214 -0.554212904 5.142403e-01 0.13131313 2 NA
## 215 -0.590720180 6.112333e-01 0.14141414 2 NA
## 216 -0.622512741 6.365401e-01 0.15151515 2 NA
## 217 -0.644557724 5.876896e-01 0.16161616 2 NA
## 218 -0.652660559 4.793831e-01 0.17171717 2 NA
## 219 -0.643842115 3.387298e-01 0.18181818 2 NA
## 220 -0.616628805 1.976362e-01 0.19191919 2 NA
## 221 -0.571221620 8.438309e-02 0.20202020 2 NA
## 222 -0.509510497 1.687328e-02 0.21212121 2 NA
## 223 -0.434793049 1.397016e-03 0.22222222 2 NA
## 224 -0.351862077 2.606923e-02 0.23232323 2 NA
## 225 -0.267682963 5.415410e-02 0.24242424 2 NA
## 226 -0.189397508 6.226882e-02 0.25252525 2 NA
## 227 -0.124076162 4.389530e-02 0.26262626 2 NA
## 228 -0.078359553 1.244745e-02 0.27272727 2 NA
## 229 -0.058812668 2.723043e-03 0.28282828 2 NA
## 230 -0.069599529 5.655520e-02 0.29292929 2 NA
## 231 -0.108806479 1.928222e-01 0.30303030 2 NA
## 232 -0.174389802 4.242541e-01 0.31313131 2 NA
## 233 -0.263246822 7.512533e-01 0.32323232 2 NA
## 234 -0.371284175 1.161259e+00 0.33333333 2 NA
## 235 -0.493719785 1.631420e+00 0.34343434 2 NA
## 236 -0.625445077 2.133115e+00 0.35353535 2 NA
## 237 -0.761383168 2.636948e+00 0.36363636 2 NA
## 238 -0.896802724 3.117013e+00 0.37373737 2 NA
## 239 -1.027561423 3.553709e+00 0.38383838 2 NA
## 240 -1.150265404 3.934874e+00 0.39393939 2 NA
## 241 -1.262342470 4.255439e+00 0.40404040 2 NA
## 242 -1.362036399 4.516096e+00 0.41414141 2 NA
## 243 -1.448336919 4.721488e+00 0.42424242 2 NA
## 244 -1.520864473 4.878403e+00 0.43434343 2 NA
## 245 -1.579730834 4.994244e+00 0.44444444 2 NA
## 246 -1.625396374 5.075925e+00 0.45454545 2 NA
## 247 -1.658542841 5.129181e+00 0.46464646 2 NA
## 248 -1.679977352 5.158202e+00 0.47474747 2 NA
## 249 -1.690579465 5.165492e+00 0.48484848 2 NA
## 250 -1.691298854 5.151837e+00 0.49494949 2 NA
## 251 -1.683206329 5.116320e+00 0.50505051 2 NA
## 252 -1.667595689 5.056352e+00 0.51515152 2 NA
## 253 -1.646128082 4.967755e+00 0.52525253 2 NA
## 254 -1.621004114 4.844953e+00 0.53535354 2 NA
## 255 -1.595142308 4.681379e+00 0.54545455 2 NA
## 256 -1.572336091 4.470207e+00 0.55555556 2 NA
## 257 -1.557356453 4.205462e+00 0.56565657 2 NA
## 258 -1.555964928 3.883504e+00 0.57575758 2 NA
## 259 -1.574803094 3.504724e+00 0.58585859 2 NA
## 260 -1.621131603 3.075130e+00 0.59595960 2 NA
## 261 -1.702404632 2.607362e+00 0.60606061 2 NA
## 262 -1.825684453 2.120666e+00 0.61616162 2 NA
## 263 -1.996924357 1.639434e+00 0.62626263 2 NA
## 264 -2.220174024 1.190279e+00 0.63636364 2 NA
## 265 -2.496786083 7.980029e-01 0.64646465 2 NA
## 266 -2.824722273 4.813207e-01 0.65656566 2 NA
## 267 -3.198069358 2.494435e-01 0.66666667 2 NA
## 268 -3.606880463 1.005985e-01 0.67676768 2 NA
## 269 -4.037482432 2.316028e-02 0.68686869 2 NA
## 270 -4.473757282 0.000000e+00 0.69696970 2 NA
## 271 -4.898867836 1.210997e-02 0.70707071 2 NA
## 272 -5.285129904 2.401856e-02 0.71717172 2 NA
## 273 -5.607138698 2.207144e-02 0.72727273 2 NA
## 274 -5.842529864 9.514019e-03 0.73737374 2 NA
## 275 -5.971199355 0.000000e+00 0.74747475 2 NA
## 276 -5.974606769 9.750020e-03 0.75757576 2 NA
## 277 -5.865435321 2.343632e-02 0.76767677 2 NA
## 278 -5.658282034 2.713664e-02 0.77777778 2 NA
## 279 -5.370632247 1.611479e-02 0.78787879 2 NA
## 280 -5.023536988 9.300061e-04 0.79797980 2 NA
## 281 -4.641575308 1.515596e-02 0.80808081 2 NA
## 282 -4.243407303 8.205219e-02 0.81818182 2 NA
## 283 -3.846041349 2.183977e-01 0.82828283 2 NA
## 284 -3.465150652 4.370326e-01 0.83838384 2 NA
## 285 -3.113406158 7.412981e-01 0.84848485 2 NA
## 286 -2.799919177 1.123494e+00 0.85858586 2 NA
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```
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